Design and Application of A Deformation Time-frequency Analysis Software Package Based on MATLAB
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摘要: 为方便台站工作人员快速准确分析定点形变观测数据的时频响应特征，利用MATLAB软件研发了基于小波分析和同步挤压时频变换的交互式数据处理与成图软件包。该软件包遵循模块化设计原则，利用导入模块读取从中国地震前兆台网数据处理系统下载的原始数据，利用预处理模块对缺失数据进行插值补全，利用小波分解与重构模块从原始数据中提取待分析的目标信号分量，利用同步挤压时频分析模块对提取目标进行高精度时频分析，并在各模块关键节点中增加绘图功能，全部处理过程采用绘图-参数输入-绘图交互的方式进行，参数灵活可调，且每步计算结果直观清晰。应用该软件包对2020年1月至2021年6月西昌小庙台DSQ型水管仪和SS-Y型伸缩仪整时值采样数据进行固体潮时频计算，并与理论固体潮时频结果进行对比，结合时频辅助分析方法，从时频谱角度初步评价了2套仪器观测资料的质量情况，为台站日常数据跟踪分析提供了参考。Abstract: In order to facilitate the station staff to effectively and accurately analyze the time-frequency characteristics of the fixed point deformation data, we provide an interactive data processing and mapping software package of wavelet decomposition and synchrosqueezing time-frequency transform based on MATLAB. The package follows the principle of modularization and includes such modules as the import module reading the original data from the txt file downloaded by QZProcess, the preprocess module interpolating the missing original data, the wavelet decomposition and reconstruction module extracting the target signal component from the original data and the synchrosqueezing transform module carrying out high precision time-frequency results of the extracted target signal. Since the mapping functions are integrated into modules and the whole flow is operated interactively, the calculation parameters are adjustable and the result of each step is intuitive. We use the proposed software package to process the hourly sampling data for DSQ and SS-Y of Xiaomiao station from Jan,2020 to June,2021. The processing results of earth tide signal qualitatively evaluate the quality of observation, and also providing useful reference for precursory data tracking and analysis.
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图 1 软件总体处理框架流程
Figure 1. General processing framework and flow of the software package
3 小庙台2020年1月至2021年6月DSQ型水管仪观测固体潮与理论固体潮时频结果对比
3. Comparison of time-frequency results between observed and theoretical earth tide of DSQ data of Xiaomiao seismic station from Jan, 2020 to June,2021
4 小庙台2020年1月至2021年6月SS-Y型伸缩仪观测固体潮与理论固体潮时频结果对比
4. Comparison of time-frequency results between observed and theoretical earth tide of SS-Y data of Xiaomiao seismic station from Jan,2020 to June,2021
表 1 软件包所用方法及基本定义
Table 1. Summary of methods and basic definitions of the software package
类型 方法名称 基本定义 备注 信号分解方法 小波分析
（DWT）${A}_{j}s\left(t\right)={A}_{j+1}s\left(t\right)+{D}_{j+1}s\left(t\right),j=\mathrm{0,1},\cdots ,N$ $ s\left(t\right) $为原始信号，$ N $为分解层数，$ {A}_{j}s\left(t\right) $为第$ j $阶低频趋势分量，$ {D}_{j}s\left(t\right) $为第$ j $阶高频细节分量 时频前处理
方法连续小波变换
（CWT）$ C W T\left(a,\tau \right)=\left\langle{s,{\psi }_{a,\tau }}\right\rangle={\displaystyle\int }_{-\infty }^{+\infty }s\left(t\right){a}^{-\frac{1}{2}}{\psi }^{*}\left(\dfrac{t-\tau }{a}\right){\rm{d}}t $ $ {a}^{-\frac{1}{2}}\psi \left(\dfrac{t-\tau }{a}\right) $为母小波族，$ a $为小波尺度因子，$ \tau $为时间平移因子 广义S变换
（GST）$ G S T\left(f,t\right)={\displaystyle\int }_{-\infty }^{+\infty }s\left(\tau \right)\dfrac{{\left|f\right|}^{\lambda }}{\sqrt{2{\text{π}} }\rho }{\rm{e}}^{-\dfrac{{\left(\tau -t\right)}^{2}{f}^{2\lambda }}{2{\rho }^{2}}}{\rm{e}}^{-j2{\text{π}} \tau }{\rm{d}}\tau $ $ \dfrac{\left|f\right|}{\sqrt{2{\text{π}} }}{\rm{e}}^{-\dfrac{{t}^{2}{f}^{2}}{2}} $为Gauss窗，$ \lambda $为时宽调节参数，$ \rho $为衰减趋势调节参数 时频后处理
方法同步挤压小波变换（SSCWT） $ S S C W T\left({\omega }_{l},\tau \right)={\Delta \omega }^{-1}\displaystyle\sum _{{a}_{k}:\left|C W T\left(a,\tau \right)-{\omega }_{l}\right|\leqslant \Delta \omega /2}C W T\left(a,\tau \right){{a}_{k}}^{-3/2}\left({\Delta a}_{k}\right) $ $ C W T\left(a,\tau \right) $为CWT时频谱，$ {a}_{k} $为第$ k $个离散化尺度，$ {\omega }_{l} $为点$ l $处离散化频率值 同步挤压广义S变换（SSGST） $ S S G S T\left({\tilde f}_{l},t\right)={(\Delta {\tilde f}_{l})}^{-1}\displaystyle\sum _{{f}_{k}:\left|{\tilde f}_{l}\left({f}_{k},t\right)-{\tilde f}_{l}\right|\leqslant \Delta {\tilde f}_{l}/2}\left|GST\left({f}_{k},t\right)\right|{f}_{k}\Delta {f}_{k} $ $ GST\left({f}_{k},t\right) $为GST时频谱，$ {f}_{k} $为离散频点，$ {\tilde f}_{l} $为挤压中心频率，$ \Delta {\tilde f}_{l} $为挤压带宽 辅助分析
方法叠加幅值特征函数（SCCF） $ S C C F\left(t\right)=\displaystyle\sum _{t=1}^{N}\left|TFR(f,t)\right|f=1,\cdots ,M $ $ TFR(f,t)\in {\boldsymbol{C}}^{M\cdot N} $，为时频谱矩阵 叠加系数包络函数（SCEF） $ S C E F\left(f\right)=\displaystyle\sum _{f=1}^{M}\left|E(f,t)\right|t=1,\cdots ,N $ $ E(f,t) $为频率点$ f $处时频谱系数的包络函数 信息熵（Renyi） $ {H}_{a}\left(T F R\right)=\dfrac{1}{1-a}{\mathrm{log}}_{2}\displaystyle\iint T F R(f,t){\rm{d}}t{\rm{d}}f $ $ a $为Renyi熵的阶次，一般取3，$ {H}_{a} $为熵值 
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表 2 软件包所含函数程序与功能
Table 2. Summary of programs and functions of the software package
函数（脚本）名称 所属功能模块 主要功能描述 Main_func.m 主脚本程序 封装全部功能模块函数，执行全计算流程 deformation_txt_read.m 数据导入模块 导入TXT文件中原始数据及测项信息 deformation_data_interp.m 数据预处理模块 利用DCT-PLS方法平滑插值缺失数据 deformation_mallat_wavelet.m 小波分解与重构模块 利用DWT分解并提取待分析目标信号 deformation_timefrequency_analysis.m 同步挤压时频分析模块 分频计算SSCWT或SSGST时频谱，并计算SCCF函数、
SCEF函数、Renyi熵等辅助分析项dctpls_smoothn.m 模块子调用函数 封装DCT-PLS插值方法实现算法 sscwt_tfrs.m 模块子调用函数 封装SSCWT时频方法实现算法 ssgst_tfrs.m 模块子调用函数 封装SSGST时频方法实现算法
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表 3 观测固体潮与理论固体潮时频谱Renyi熵值
Table 3. The Renyi entropy of observed earth tide and theoretical earth tide
项目 DSQ（NS向） DSQ（EW向） SS-Y（NS向） SS-Y（EW向） 观测固体潮时频谱Renyi熵值 6.26 5.56 8.44 8.17 理论固体潮时频谱Renyi熵值 5.50 5.50 5.13 5.13 Renyi熵误差 0.76 0.06 3.31 3.04
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